WEAK CONVERGENCE OF SEQUENCES OF FIRST PASSAGE PROCESSES AND APPLICATIONS

2012 ◽  
Author(s):  
STEFAN S. RALESCU ◽  
MADAN L. PURI
Keyword(s):  
Weak Convergence ◽  
First Passage ◽  
Convergence Of Sequences

An invariance principle in k-dimensional extended renewal theory

1979 ◽  
Vol 16 (3) ◽  
pp. 567-574 ◽  
Author(s):  
Attila Csenki
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Covariance Matrix ◽  
Invariance Principle ◽  
Exit Time ◽  
Renewal Theory ◽  
Random Vectors ◽  
First Exit Time ◽  
First Passage ◽  
Passage Times

Let ·be a sequence of k -dimensional i.i.d. random vectors and define the first-passage times for where (cvτ)v, τ= 1,· ··,k is the covariance matrix of In this paper the weak convergence of Zn in (D[0, ∞))k is proved under the assumption (0,∞) for all v = 1, ···, k. We deduce the result from the Donsker invariance principle by means of Theorem 5.5 of Billingsley (1968). This method is also used to derive a limit theorem for the first-exit time Mn = min{Nnt for fixed t1,···, tk > 0. The second result is an extension of a theorem of Hunter (1974) whose method of proof applies only if Ρ (ξ1 [0,∞)k) = 1 and μ ν = tv for all v = 1, ···, k.

Weak convergence of sequences of semimartingales with applications to multitype branching processes

1986 ◽  
Vol 18 (1) ◽  
pp. 20-65 ◽  
Author(s):  
A. Joffe ◽  
M. Metivier
Keyword(s):  
Weak Convergence ◽  
Stochastic Processes ◽  
Special Class ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Convergence Of Sequences ◽  
Multitype Branching Processes

The paper is devoted to a systematic discussion of recently developed techniques for the study of weak convergence of sequences of stochastic processes. The methods described make essential use of the semimartingale structure of the processes. Sufficient conditions for tightness including the results of Rebolledo are derived. The techniques are applied to a special class of processes, namely the D-semimartingales. Applications to multitype branching processes are given.

Weak convergence of sequences of semimartingales with applications to multitype branching processes

1986 ◽  
Vol 18 (01) ◽  
pp. 20-65 ◽  
Author(s):  
A. Joffe ◽  
M. Metivier
Keyword(s):  
Weak Convergence ◽  
Stochastic Processes ◽  
Special Class ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Convergence Of Sequences ◽  
Multitype Branching Processes

The paper is devoted to a systematic discussion of recently developed techniques for the study of weak convergence of sequences of stochastic processes. The methods described make essential use of the semimartingale structure of the processes. Sufficient conditions for tightness including the results of Rebolledo are derived. The techniques are applied to a special class of processes, namely the D-semimartingales. Applications to multitype branching processes are given.

Weak convergence of first passage time processes

1971 ◽  
Vol 8 (2) ◽  
pp. 417-422 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Stochastic Process ◽  
Weak Convergence ◽  
First Passage Time ◽  
Continuous Mapping ◽  
Passage Time ◽  
Continuous Functions ◽  
First Passage ◽  
Image Position ◽  
Supremum Process

Let D = D[0, ∞) be the space of all real-valued right-continuous functions on [0, ∞) with limits from the left. For any stochastic process X in D, let the associated supremum process be S(X), wherefor any x ∊ D. It is easy to verify that S: D → D is continuous in any of Skorohod's (1956) topologies extended from D[0,1] to D[0, ∞) (cf. Stone (1963) and Whitt (1970a, c)). Hence, weak convergence Xn ⇒ X in D implies weak convergence S(Xn) ⇒ S(X) in D by virtue of the continuous mapping theorem (cf. Theorem 5.1 of Billingsley (1968)).

Weak convergence of sequences of random elements with random indices

1980 ◽  
Vol 88 (1) ◽  
pp. 171-174 ◽  
Author(s):  
M. Csörgő ◽  
Z. Rychlik
Keyword(s):  
Positive Integer ◽  
Weak Convergence ◽  
Metric Space ◽  
Probability Space ◽  
Random Element ◽  
Random Variables ◽  
Random Elements ◽  
Convergence Of Sequences ◽  
Random Indices ◽  
Separable Metric Space

Let (S, d) be a separable metric space equipped with its Borel σ field . Let {Yn, n ≥ 1} be a sequence of S-valued random elements defined on a probability space (Ω, , p). Assume Yn ⇒ Y converges weakly to an S-valued random element Y. Let {Nn, n ≥ 1} be a sequence of positive integer-valued random variables defined on the same probability space (Ω, , p).

On the weak convergence of sequences of circuit processes: a probabilistic approach

1992 ◽  
Vol 29 (02) ◽  
pp. 374-383 ◽  
Author(s):  
Sophia Kalpazidou
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Asymptotic Behaviour ◽  
Markov Processes ◽  
Probabilistic Approach ◽  
Probabilistic Interpretation ◽  
Sample Paths ◽  
Convergence Of Sequences ◽  
Recurrent Markov Chain ◽  
Positive Recurrent

The asymptotic behaviour of sequences of Markov processes whose finite distributions depend upon the sample paths ω of a positive recurrent Markov chain ξ is studied. The existence of such sequences depends upon the existence of a unique class of directed weighted circuits having a probabilistic interpretation in terms of the directed circuits occurring along the sample paths of ξ. An application to multiple Markov chains is given.

The time constant of first-passage percolation on the square lattice

1980 ◽  
Vol 12 (4) ◽  
pp. 864-879 ◽  
Author(s):  
J. Theodore Cox
Keyword(s):  
Distribution Function ◽  
Weak Convergence ◽  
Time Constant ◽  
Square Lattice ◽  
First Passage Percolation ◽  
First Passage ◽  
Underlying Distribution

Let μ (F) be the time constant of first-passage percolation on the square lattice with underlying distribution function F. Two theorems are presented which show, under some restrictions, that μ varies continuously in F with respect to weak convergence. These results are improvements of existing continuity theorems.

Sign in / Sign up

Export Citation Format

Share Document